Prove that $(z^3-z)(z+2)$ is divisible by $12$ for all integers $z$

Question

I am a student and this question is part of my homework.

May you tell me if my proof is correct?

Thanks for your help!

Prove that $(z^3-z)(z+2)$ is divisible by $12$ for all integers $z$.

$(z^3-z)(z+2)=z(z^2-1)(z+2)=z(z-1)(z+1)(z+2)=(z-1)(z)(z+1)(z+2)$

$(z-1)(z)(z+1)(z+2)$ means the product of $4$ consecutive numbers.

Any set of $4$ consecutive numbers has $2$ even numbers, then $(z-1)(z)(z+1)(z+2)$ is divisible by $4$.

Any set of $4$ consecutive number has at least one number that is multiple of $3$, then $(z-1)(z)(z+1)(z+2)$ is divisible by 3.

Therefore $(z-1)(z)(z+1)(z+2)$ is divisible by $12$. Q.E.D.

You should use the term "integer" instead of number. And you can actually prove something stonger - that it is divisible by $24$. Your proof for $12$ looks correct, though. — Thomas Andrews, Feb 12 '15 at 03:42
@ThomasAndrews How can you prove that is divisible by 24? Thanks! — Beginner, Feb 12 '15 at 03:59
You can prove it by first proving that the product of three consecutive numbers is divisible by $6$, then prove it by induction on $x$. (That only proves it for $x\geq1$, but it follows easily that it is also true for $x\leq 0$.) — Thomas Andrews, Feb 12 '15 at 04:07
@Beginner. See http://math.stackexchange.com/questions/12067/the-product-of-n-consecutive-integers-is-divisible-by-n-without-using-the-prop — lab bhattacharjee, Feb 12 '15 at 04:40

score 7 · Accepted Answer · answered Feb 12 '15 at 05:16

7

That's correct. Alternatively it is divisible by $\,24\,$ by integrality of binomial coefficients

$$\,(z+2)(z+1)z(z-1)\, =\, 4!\ \dfrac{(z+2)(z+1)z(z-1)}{4!}\, =\, 24{ {z+2\choose 4}}\qquad\qquad$$

Similarly $\,n!\,$ divides the product of $\,n\,$ consecutive naturals.

answered Feb 12 '15 at 05:16

Bill Dubuque

@billdubusque Would you mind to explain the last step? I used the binomial and factorial formulas and I got a different result. But, I really like the idea. I did not think about that possibility. Thanks! – Beginner Feb 12 '15 at 14:37
@Beginner $$ {z+2 \choose 4} ,=, \frac{(z+2)!}{4!,(z-2)!}, =, \frac{(z+2)(z+1)z(z-1)}{4!}$$ – Bill Dubuque Feb 12 '15 at 15:05
I had a mistake in my computations. Thanks for your help! – Beginner Feb 12 '15 at 15:29

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